Tube Category, Tensor Renormalization and Topological Holography

Abstract

Ocneanu's tube algebra provides a finite algorithm to compute the Drinfeld center of a fusion category. In this work we reveal the universal property underlying the tube algebra. Take a base category $\mathcal V$ which is strongly concrete, bicomplete, and closed symmetric monoidal. For physical applications one takes $\mathcal V=\mathbf{Vect}$ the category of vector spaces. Given a $\mathcal V$-enriched rigid monoidal category $\mathcal C$ (not necessarily finite or semisimple) we define the tube category $\mathbb X \mathcal C$ using coends valued in $\mathcal V$. Our main theorem established the relation between (the category of representations of) the tube category $\mathbb X \mathcal C$ and the Drinfeld center $Z(\mathcal C)$: $Z(\mathcal C)\hookrightarrow \mathrm{Fun}(\mathbb X \mathcal C^{\mathrm{op}},\mathcal V)\cong Z(\mathcal C\hookrightarrow\mathrm{Fun}(\mathcal C^{\mathrm{op}},\mathcal V))\hookrightarrow Z(\mathrm{Fun}(\mathcal C^{\mathrm{op}},\mathcal V))$. Physically, besides viewing the tube category as a version of TFT with domain being the tube, we emphasize the "Wick-rotated" perspective, that the morphisms in $\mathbb X \mathcal C$ are the local tensors of fixed-point matrix product operators which preserves the symmetry $\mathcal C$ in one spatial dimension. We provide a first-principle flavored construction, from microscopic quantum degrees of freedom and operators preserving the symmetry, to the macroscopic universal properties of the symmetry which form the Drinfeld center. Our work is thus a proof to the 1+1D topological holography in a very general setting.